Finite Group Databasebeta

← #111550

Subgroups — #111550

Order 512.

1194 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic
2#21818C211yesmaximal, characteristic
3C2×#1479C212yesmaximal
4C2×#1461C211yesmaximal, characteristic
5C2×#1276C212yesmaximal
6C22×C4⋊Q8C411yescharacteristic
7C22×C4×Q8C2211yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#431C2212yes—
10C2×#431C2212yes—
11C4×C4⋊Q8—24no—
12C4×C4⋊Q8—24no—
13C2×#391C411yescharacteristic
14C2×#391C411yescharacteristic
15C2×#389—22no—
16C2×#384C411yescharacteristic
17C2×C4×C4⋊C4—22no—
18C2×C43C2211yescharacteristic
19#1984—24no—
20#1817—22no—
21#1479—24no—
22#1479—24no—
23C2×Q8⋊C8—24no—
24#1461—24no—
25C2×#324—22no—
26#1276—24no—
27C23×Q8C2×C411yescharacteristic
28C23×Q8C2×C411yescharacteristic
29C2×C4⋊Q8—22no—
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8C814yescomplemented
35C2×C4×Q8—22no—
36C2×C4×Q8—22no—
37C2×C4×Q8—22no—
38C2×C4×Q8—22no—
39C2×C4×Q8—22no—
40C2×C4×Q8—22no—
41C2×C4×Q8—22no—
42C2×C4×Q8—22no—
43C2×C4×Q8—22no—
44C2×C4×Q8—22no—
45C2×C4×Q8—22no—
46C2×C4×Q8D414yes—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4D412yes—
53C22×C42—21no—
54C22×C42—21no—
55C22×C42—21no—
56C22×C42C2×C411yescharacteristic
57C22×C42C2×C411yescharacteristic
58C22×C42D412yes—
59C22×C42C2311yescharacteristic
60#431—22no—
61#431—22no—
62#431—22no—
63#431—22no—
64#431—22no—
65#431—22no—
66#431—22no—
67#431D414yes—
68C2×C4⋊C8—22no—
69C2×C4⋊C8D414yes—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C4×C8—22no—
73C2×C4×C8—22no—
74#391—22no—
75#391—22no—
76#391—22no—
77#391—22no—
78#389—216no—
79#384—22no—
80#384—22no—
81C4×C4⋊C4—216no—
82C2×#95—22no—
83C2×#95—22no—
84C43—22no—
85C43D414yes—
86#336—28no—
87Q8⋊C8—216no—
88#324—28no—
89C22×Q8—21no—
90C22×Q8—21no—
91C22×Q8—21no—
92C22×Q8—22no—
93C22×Q8—22no—
94C22×Q8—21no—
95C22×Q8—21no—
96C22×Q8—21no—
97C22×Q8—21no—
98C22×Q8—22no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8—22no—
102C22×Q8—22no—
103C22×Q8—22no—
104C22×Q8C22⋊C412yes—
105C22×Q8C2×C812yes—
106C22×Q8C2×C812yes—
107C22×Q8M1612yes—
108C22×Q8M1612yes—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4C22⋊C412yes—
114C23×C4C22⋊C412yes—
115C23×C4C22×C411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C22×C8—24no—
119C22×C8—42no—
120C4⋊Q8—28no—
121C4⋊Q8—28no—
122C4⋊Q8—28no—
123C4⋊Q8—28no—
124C4×Q8—24no—
125C4×Q8—24no—
126C4×Q8—24no—
127C4×Q8—24no—
128C4×Q8—42no—
129C4×Q8—42no—
130C4×Q8—42no—
131C4×Q8—42no—
132C4×Q8—42no—
133C4×Q8—42no—
134C4×Q8—42no—
135C4×Q8—42no—
136C4×Q8—42no—
137C4×Q8—42no—
138C4×Q8—42no—
139C4×Q8—42no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—48no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4C22⋊C412yes—
161C2×C42—28no—
162C2×C42—21no—
163C2×C42—24no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—21no—
169C2×C42—24no—
170C2×C42—24no—
171C2×C42—24no—
172C2×C42—22no—
173C2×C42—24no—
174C2×C42—22no—
175C2×C42—44no—
176C2×C42—41no—
177C2×C42—42no—
178C2×C42—41no—
179C2×C42C22⋊C412yes—
180C2×C42C22⋊C412yes—
181C2×C42C2×C812yes—
182C2×C42M1612yes—
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42#4112yes—
188C2×C42#4112yes—
189C4⋊C8—24no—
190C4⋊C8—24no—
191C4⋊C8—24no—
192C4⋊C8—42no—
193C22⋊C8—24no—
194C22⋊C8—42no—
195C22⋊C8—42no—
196C22⋊C8—42no—
197C4×C8—24no—
198C4×C8—42no—
199C4×C8—42no—
200C4×C8—42no—
201#95—28no—
202#95—28no—
203C24C2×C22⋊C411yescharacteristic
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—24no—
212C2×Q8—24no—
213C2×Q8—24no—
214C2×Q8—24no—
215C2×Q8—24no—
216C2×Q8—24no—
217C2×Q8—24no—
218C2×Q8—24no—
219C2×Q8—24no—
220C2×Q8—24no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8C22⋊C814yes—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4C22⋊C814yes—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C4×D412yes—
273C22×C4C4×D412yes—
274C22×C4#12012yes—
275C22×C4#12011yescharacteristic
276C22×C4#12012yes—
277C22×C4#12011yescharacteristic
278C22×C4#12112yes—
279C22×C4#12112yes—
280C22×C4#12311yescharacteristic
281C22×C4#12311yescharacteristic
282C22×C4C22×C811yescharacteristic
283C22×C4C2×M1611yescharacteristic
284C22×C4C2×M1611yescharacteristic
285C22×C4C2×M1611yescharacteristic
286C22×C4#13112yes—
287C22×C4#13112yes—
288C2×C8—44no—
289C2×C8—44no—
290C2×C8—48no—
291C2×C8—84no—
292C4⋊C4—232no—
293C4⋊C4—24no—
294C4⋊C4—416no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C42—216no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—24no—
307C42—48no—
308C42—44no—
309C42—44no—
310C42—44no—
311C42—44no—
312C42—42no—
313C42C22⋊C814yes—
314C42#12012yes—
315C42#12112yes—
316C42#12112yes—
317C42#12112yes—
318C42#12112yes—
319C42#12312yes—
320C42#12412yes—
321C42#12712yes—
322C23—22no—
323C23—22no—
324C23#37911yescharacteristic
325C23#38611yescharacteristic
326C23#38611yescharacteristic
327C23C2×C22⋊C811yescharacteristic
328C23#40711yescharacteristic
329C23#40811yescharacteristic
330C23#40811yescharacteristic
331Q8—28no—
332Q8—44no—
333Q8—44no—
334Q8—44no—
335Q8—44no—
336Q8—44no—
337Q8—44no—
338Q8—44no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—28no—
342C2×C4—28no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4#38612yes—
379C2×C4#39012yes—
380C2×C4#39212yes—
381C2×C4#39212yes—
382C2×C4#39412yes—
383C2×C4#39412yes—
384C2×C4C2×C22⋊C811yescharacteristic
385C2×C4C2×C22⋊C811yescharacteristic
386C2×C4#40711yescharacteristic
387C2×C4#40711yescharacteristic
388C2×C4#40812yes—
389C2×C4#40812yes—
390C2×C4C8×D412yes—
391C2×C4C8×D412yes—
392C2×C4#43512yes—
393C2×C4#43512yes—
394C2×C4#43512yes—
395C2×C4#43512yes—
396C2×C4#43612yes—
397C2×C4#43612yes—
398C2×C4#45614yes—
399C8—84no—
400C8—84no—
401C22—22no—
402C22—22no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—22no—
408C22—22no—
409C22#143811yescharacteristic
410C22#177911yescharacteristic
411C22#178011yescharacteristic
412C22#186911yescharacteristic
413C22#186911yescharacteristic
414C22#187011yescharacteristic
415C22#187011yescharacteristic
416C22#188412yes—
417C22#188612yes—
418C4—28no—
419C4—22no—
420C4—22no—
421C4—22no—
422C4—44no—
423C4—44no—
424C4—44no—
425C4—82no—
426C4—82no—
427C4#186912yes—
428C4#187012yes—
429C4#195412yes—
430C4#195512yes—
431C4#200514yes—
432C2—21no—
433C2—21no—
434C2—21no—
435C2—21no—
436C2#895812yes—
437C2#896212yes—
438C2#1096411yescharacteristic
439C2#1097811yescharacteristic
440C2#1097911yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
34. C2×C4⋊Q8C8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C4×C4⋊Q82
#218182
C2×#14792
C2×#14612
C2×#12762